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The product-moment correlation coefficient and linear regression for truncated data. (English) Zbl 0882.62054
Summary: The random truncation model has been considered extensively in the literature. W. Y. Tsai [Biometrika 77, No 1, 169-177 (1990; Zbl 0692.62045)] has noted that many previous results hold under the weaker assumption of quasi-independence between the failure time and the truncation time in the observable region of truncated data. We generalize the Pearson product-moment correlation coefficient to measure the association between both time variables in the observable region. We show that if the failure time and the truncation time follow a truncated bivariate normal distribution, then a zero value of the generalized correlation coefficient is equivalent to the quasi-independence.
We propose a corresponding sample correlation coefficient and consider its asymptotic behavior. We also study an application of quasi-independence to truncated linear regression with its asymptotic results. The proposed estimator, stemming directly from the least-squares approach, is computationally much simpler and has a natural extension to multiple linear regression. A simulation study shows that the proposed estimator for regression slope competes well with available nonparametric estimators.

MSC:
62J05 Linear regression; mixed models
62H20 Measures of association (correlation, canonical correlation, etc.)
62G07 Density estimation
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